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Theorem kqreglem1 21354
 Description: A Kolmogorov quotient of a regular space is regular. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
Assertion
Ref Expression
kqreglem1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Reg)
Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem kqreglem1
Dummy variables 𝑚 𝑤 𝑧 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 kqval.2 . . . . 5 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
21kqtopon 21340 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
32adantr 480 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
4 topontop 20541 . . 3 ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → (KQ‘𝐽) ∈ Top)
53, 4syl 17 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Top)
6 toponss 20544 . . . . . . . 8 (((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) ∧ 𝑎 ∈ (KQ‘𝐽)) → 𝑎 ⊆ ran 𝐹)
73, 6sylan 487 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) → 𝑎 ⊆ ran 𝐹)
87sselda 3568 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏𝑎) → 𝑏 ∈ ran 𝐹)
91kqffn 21338 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
109ad3antrrr 762 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏𝑎) → 𝐹 Fn 𝑋)
11 fvelrnb 6153 . . . . . . 7 (𝐹 Fn 𝑋 → (𝑏 ∈ ran 𝐹 ↔ ∃𝑧𝑋 (𝐹𝑧) = 𝑏))
1210, 11syl 17 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏𝑎) → (𝑏 ∈ ran 𝐹 ↔ ∃𝑧𝑋 (𝐹𝑧) = 𝑏))
138, 12mpbid 221 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏𝑎) → ∃𝑧𝑋 (𝐹𝑧) = 𝑏)
14 simpllr 795 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) → 𝐽 ∈ Reg)
151kqid 21341 . . . . . . . . . . . . . . 15 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
1615ad3antrrr 762 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
17 simplr 788 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) → 𝑎 ∈ (KQ‘𝐽))
18 cnima 20879 . . . . . . . . . . . . . 14 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑎 ∈ (KQ‘𝐽)) → (𝐹𝑎) ∈ 𝐽)
1916, 17, 18syl2anc 691 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) → (𝐹𝑎) ∈ 𝐽)
209adantr 480 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → 𝐹 Fn 𝑋)
2120adantr 480 . . . . . . . . . . . . . . 15 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) → 𝐹 Fn 𝑋)
22 elpreima 6245 . . . . . . . . . . . . . . 15 (𝐹 Fn 𝑋 → (𝑧 ∈ (𝐹𝑎) ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)))
2321, 22syl 17 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) → (𝑧 ∈ (𝐹𝑎) ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)))
2423biimpar 501 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) → 𝑧 ∈ (𝐹𝑎))
25 regsep 20948 . . . . . . . . . . . . 13 ((𝐽 ∈ Reg ∧ (𝐹𝑎) ∈ 𝐽𝑧 ∈ (𝐹𝑎)) → ∃𝑤𝐽 (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))
2614, 19, 24, 25syl3anc 1318 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) → ∃𝑤𝐽 (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))
27 simp-4l 802 . . . . . . . . . . . . . 14 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → 𝐽 ∈ (TopOn‘𝑋))
28 simprl 790 . . . . . . . . . . . . . 14 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → 𝑤𝐽)
291kqopn 21347 . . . . . . . . . . . . . 14 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) → (𝐹𝑤) ∈ (KQ‘𝐽))
3027, 28, 29syl2anc 691 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → (𝐹𝑤) ∈ (KQ‘𝐽))
31 simprrl 800 . . . . . . . . . . . . . 14 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → 𝑧𝑤)
32 simplrl 796 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → 𝑧𝑋)
331kqfvima 21343 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽𝑧𝑋) → (𝑧𝑤 ↔ (𝐹𝑧) ∈ (𝐹𝑤)))
3427, 28, 32, 33syl3anc 1318 . . . . . . . . . . . . . 14 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → (𝑧𝑤 ↔ (𝐹𝑧) ∈ (𝐹𝑤)))
3531, 34mpbid 221 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → (𝐹𝑧) ∈ (𝐹𝑤))
36 topontop 20541 . . . . . . . . . . . . . . . . . 18 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3727, 36syl 17 . . . . . . . . . . . . . . . . 17 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → 𝐽 ∈ Top)
38 elssuni 4403 . . . . . . . . . . . . . . . . . 18 (𝑤𝐽𝑤 𝐽)
3938ad2antrl 760 . . . . . . . . . . . . . . . . 17 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → 𝑤 𝐽)
40 eqid 2610 . . . . . . . . . . . . . . . . . 18 𝐽 = 𝐽
4140clscld 20661 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑤 𝐽) → ((cls‘𝐽)‘𝑤) ∈ (Clsd‘𝐽))
4237, 39, 41syl2anc 691 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → ((cls‘𝐽)‘𝑤) ∈ (Clsd‘𝐽))
431kqcld 21348 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘𝑤) ∈ (Clsd‘𝐽)) → (𝐹 “ ((cls‘𝐽)‘𝑤)) ∈ (Clsd‘(KQ‘𝐽)))
4427, 42, 43syl2anc 691 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → (𝐹 “ ((cls‘𝐽)‘𝑤)) ∈ (Clsd‘(KQ‘𝐽)))
4540sscls 20670 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑤 𝐽) → 𝑤 ⊆ ((cls‘𝐽)‘𝑤))
4637, 39, 45syl2anc 691 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → 𝑤 ⊆ ((cls‘𝐽)‘𝑤))
47 imass2 5420 . . . . . . . . . . . . . . . 16 (𝑤 ⊆ ((cls‘𝐽)‘𝑤) → (𝐹𝑤) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑤)))
4846, 47syl 17 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → (𝐹𝑤) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑤)))
49 eqid 2610 . . . . . . . . . . . . . . . 16 (KQ‘𝐽) = (KQ‘𝐽)
5049clsss2 20686 . . . . . . . . . . . . . . 15 (((𝐹 “ ((cls‘𝐽)‘𝑤)) ∈ (Clsd‘(KQ‘𝐽)) ∧ (𝐹𝑤) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑤))) → ((cls‘(KQ‘𝐽))‘(𝐹𝑤)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑤)))
5144, 48, 50syl2anc 691 . . . . . . . . . . . . . 14 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → ((cls‘(KQ‘𝐽))‘(𝐹𝑤)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑤)))
5220ad3antrrr 762 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → 𝐹 Fn 𝑋)
53 fnfun 5902 . . . . . . . . . . . . . . . 16 (𝐹 Fn 𝑋 → Fun 𝐹)
5452, 53syl 17 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → Fun 𝐹)
55 simprrr 801 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎))
56 funimass2 5886 . . . . . . . . . . . . . . 15 ((Fun 𝐹 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)) → (𝐹 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑎)
5754, 55, 56syl2anc 691 . . . . . . . . . . . . . 14 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → (𝐹 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑎)
5851, 57sstrd 3578 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → ((cls‘(KQ‘𝐽))‘(𝐹𝑤)) ⊆ 𝑎)
59 eleq2 2677 . . . . . . . . . . . . . . 15 (𝑚 = (𝐹𝑤) → ((𝐹𝑧) ∈ 𝑚 ↔ (𝐹𝑧) ∈ (𝐹𝑤)))
60 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑚 = (𝐹𝑤) → ((cls‘(KQ‘𝐽))‘𝑚) = ((cls‘(KQ‘𝐽))‘(𝐹𝑤)))
6160sseq1d 3595 . . . . . . . . . . . . . . 15 (𝑚 = (𝐹𝑤) → (((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎 ↔ ((cls‘(KQ‘𝐽))‘(𝐹𝑤)) ⊆ 𝑎))
6259, 61anbi12d 743 . . . . . . . . . . . . . 14 (𝑚 = (𝐹𝑤) → (((𝐹𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎) ↔ ((𝐹𝑧) ∈ (𝐹𝑤) ∧ ((cls‘(KQ‘𝐽))‘(𝐹𝑤)) ⊆ 𝑎)))
6362rspcev 3282 . . . . . . . . . . . . 13 (((𝐹𝑤) ∈ (KQ‘𝐽) ∧ ((𝐹𝑧) ∈ (𝐹𝑤) ∧ ((cls‘(KQ‘𝐽))‘(𝐹𝑤)) ⊆ 𝑎)) → ∃𝑚 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))
6430, 35, 58, 63syl12anc 1316 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → ∃𝑚 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))
6526, 64rexlimddv 3017 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) → ∃𝑚 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))
6665expr 641 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ 𝑎 → ∃𝑚 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
67 eleq1 2676 . . . . . . . . . . 11 ((𝐹𝑧) = 𝑏 → ((𝐹𝑧) ∈ 𝑎𝑏𝑎))
68 eleq1 2676 . . . . . . . . . . . . 13 ((𝐹𝑧) = 𝑏 → ((𝐹𝑧) ∈ 𝑚𝑏𝑚))
6968anbi1d 737 . . . . . . . . . . . 12 ((𝐹𝑧) = 𝑏 → (((𝐹𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎) ↔ (𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
7069rexbidv 3034 . . . . . . . . . . 11 ((𝐹𝑧) = 𝑏 → (∃𝑚 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎) ↔ ∃𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
7167, 70imbi12d 333 . . . . . . . . . 10 ((𝐹𝑧) = 𝑏 → (((𝐹𝑧) ∈ 𝑎 → ∃𝑚 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)) ↔ (𝑏𝑎 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))))
7266, 71syl5ibcom 234 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑧𝑋) → ((𝐹𝑧) = 𝑏 → (𝑏𝑎 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))))
7372com23 84 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑧𝑋) → (𝑏𝑎 → ((𝐹𝑧) = 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))))
7473imp 444 . . . . . . 7 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑧𝑋) ∧ 𝑏𝑎) → ((𝐹𝑧) = 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
7574an32s 842 . . . . . 6 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏𝑎) ∧ 𝑧𝑋) → ((𝐹𝑧) = 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
7675rexlimdva 3013 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏𝑎) → (∃𝑧𝑋 (𝐹𝑧) = 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
7713, 76mpd 15 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏𝑎) → ∃𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))
7877anasss 677 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑎 ∈ (KQ‘𝐽) ∧ 𝑏𝑎)) → ∃𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))
7978ralrimivva 2954 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → ∀𝑎 ∈ (KQ‘𝐽)∀𝑏𝑎𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))
80 isreg 20946 . 2 ((KQ‘𝐽) ∈ Reg ↔ ((KQ‘𝐽) ∈ Top ∧ ∀𝑎 ∈ (KQ‘𝐽)∀𝑏𝑎𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
815, 79, 80sylanbrc 695 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Reg)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900   ⊆ wss 3540  ∪ cuni 4372   ↦ cmpt 4643  ◡ccnv 5037  ran crn 5039   “ cima 5041  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  Topctop 20517  TopOnctopon 20518  Clsdccld 20630  clsccl 20632   Cn ccn 20838  Regcreg 20923  KQckq 21306 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-qtop 15990  df-top 20521  df-topon 20523  df-cld 20633  df-cls 20635  df-cn 20841  df-reg 20930  df-kq 21307 This theorem is referenced by:  kqreg  21364
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